Nonconvex quadratically constrained quadratic programming: best D.C. decompositions and their SDP representations

Abstract

Nonconvex quadratically constrained quadratic programming, Optimal D.C. decomposition, Semidefinite program, Piecewise linear approximation, Feasible solution,

Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2

“…. , M} using a difference of convex (D.C.) underestimators [22,87,118]; we refer to these as αBB underestimators because they specialise the generic results of Floudas and co-workers to MIQCQP [5,6,9,65]. Anstreicher [12] has shown that, no matter the choice of α parameter, a D.C. relaxation of MIQCQP is dominated by a relaxation combining McCormick [71] envelopes and a semidefinite condition.…”

“…Anstreicher [12] has shown that, no matter the choice of α parameter, a D.C. relaxation of MIQCQP is dominated by a relaxation combining McCormick [71] envelopes and a semidefinite condition. Based on this result [12], we avoid extensive computation generating the α parameters (e.g., we do not solve an LP as proposed by Zheng et al [118]). Section 4.6 presents the proposed αBB convexifications and demonstrates their importance to the GloMIQO 2 cutting plane strategy; generating αBB cuts is less computationally demanding than, for example, deriving vertex polyhedral cuts.…”

Faster, but weaker, relaxations for quadratically constrained quadratic programs

“…Closely related approaches have recently been developed by Saxena et al [17] and Zheng et al [20]. In [17], the authors study the relaxation obtained by the following spectral splitting of A i :…”

“…The paper [20] employs similar ideas but further solves a secondary SDP over different splittings of A i to improve the resultant SOCP relaxation quality.…”

“…Others have studied different types of relaxations, for example, ones based on linear programming [10,13,18] or second-order cone programming (SOCP) [9,17,20]. Generally speaking, one would expect such relaxations to provide weaker bounds in less time compared to SDP relaxations.…”